Discrete analogues of the Dixmier operators

TL;DR

This paper constructs discrete analogues of the Dixmier operators, which are commuting difference operators with polynomial coefficients related to a genus 1 spectral curve, advancing the understanding of discrete integrable systems.

Contribution

It introduces a novel construction of discrete analogues of the Dixmier operators with polynomial coefficients linked to elliptic spectral curves.

Findings

Discrete analogues of Dixmier operators are explicitly constructed.

The operators commute and correspond to genus 1 spectral curves.

This work extends the theory of integrable difference operators.

Abstract

We construct discrete analogues of the Dixmier operators, that is, commuting difference operators corresponding to a spectral curve of genus 1 whose coefficients are polynomials of the discrete variable.